How to Calculate the Surface and Lateral Area of a Prism with Formulas and Examples
The concept of Surface and Lateral Area of 3D Figures: Prisms Formulas is essential in Geometry and Mensuration, especially for students preparing for school exams, competitive exams like JEE, and for those curious about real-world uses such as design and architecture. Understanding these formulas makes it much easier to solve problems related to packaging, construction, and many other daily scenarios.
What is Surface and Lateral Area of Prisms?
A prism is a three-dimensional (3D) solid that has two identical parallel bases and flat faces connecting the bases. Examples include rectangular prisms (boxes), triangular prisms (roof shapes), and hexagonal prisms. The surface area of a prism tells us how much material would be needed to cover it entirely, while the lateral area refers only to the area of the sides, excluding the bases.
At Vedantu, we help you master these concepts so that you can quickly spot which formula to use, understand the meaning of each part of the formula, and apply your knowledge confidently.
Key Terms and Definitions
- Prism: 3D object with two parallel, congruent bases and lateral faces that are parallelograms.
- Base (B): The repeating, parallel face at the top and bottom.
- Height (h): The perpendicular distance between the two bases.
- Lateral faces: The side faces connecting the bases.
- Lateral Surface Area (LSA): Total area of only the side (lateral) faces.
- Total Surface Area (TSA): Sum of all faces’ areas — side faces plus the two bases.
- Perimeter (P): Total distance around the base.
Formulas for Surface Area of Prisms
There are two main formulas related to surface area for prisms:
- Lateral Surface Area (LSA) = Perimeter of base × Height = P × h
- Total Surface Area (TSA) = Lateral Surface Area + 2 × Area of Base = P × h + 2B
| Prism Type | Base Area (B) | Base Perimeter (P) | LSA Formula | TSA Formula |
|---|---|---|---|---|
| Rectangular Prism | l × w | 2(l + w) | 2(l + w) × h | 2(lw + lh + wh) |
| Triangular Prism | ½ × base × height | Sum of three sides of triangle | (Sum of sides) × h | (Sum of sides) × h + 2 × (area of base) |
| Hexagonal Prism | \(\frac{3\sqrt{3}}{2} a^2\) | 6a | 6a × h | 6a × h + 2 × \(\frac{3\sqrt{3}}{2} a^2\) |
Example Problems: Step-by-Step Solutions
Example 1: Rectangular Prism
Find the lateral and total surface area of a rectangular prism with length 8 cm, width 5 cm, and height 4 cm.
- Perimeter of base, P = 2(8 + 5) = 2(13) = 26 cm
- Area of base, B = 8 × 5 = 40 cm²
- Lateral Surface Area = P × h = 26 × 4 = 104 cm²
- Total Surface Area = (P × h) + 2B = 104 + 2 × 40 = 184 cm²
Example 2: Triangular Prism
A triangular prism has a base triangle with sides 3 cm, 4 cm, and 5 cm, and the length (height of prism) is 10 cm.
- Perimeter of base, P = 3 + 4 + 5 = 12 cm
- Area of base, B = ½ × 3 × 4 = 6 cm²
- Lateral Surface Area = P × h = 12 × 10 = 120 cm²
- Total Surface Area = (P × h) + 2B = 120 + 2 × 6 = 132 cm²
Practice Problems
- Find the TSA and LSA of a rectangular prism with l = 6 cm, w = 4 cm, h = 3 cm.
- A triangular prism has a base area of 10 cm², perimeter 12 cm, and height 7 cm. Find its LSA and TSA.
- Find the TSA of a hexagonal prism with base edge a = 2 cm and height h = 5 cm. (Use area formula for hexagon: \(\frac{3\sqrt{3}}{2} a^2\))
- If the base of a prism is a pentagon with area 22 cm², perimeter 15 cm, and the height is 8 cm, calculate the lateral and total surface area.
Common Mistakes to Avoid
- Mixing up lateral area with total surface area (always remember TSA includes the two bases too).
- Using height of the base (triangle) instead of the height of the prism (distance between bases).
- Forgetting to double the base area when calculating TSA.
- Mixing up perimeter and area in the formulas.
- Using inconsistent units — always stick to one unit system in each problem.
Real-World Applications
Surface and lateral area concepts are everywhere! Architects use these calculations to estimate paint or tiles for buildings. Designers compute box surface area for packaging. In science projects, knowing the area of a prism-shaped tank helps decide waterproofing material needed. Even wrapping a birthday present requires knowing the total surface area!
For more on visualizing 3D shapes, see our guide on Three-Dimensional Shapes and Their Properties or on formulas for other shapes like Surface Area of a Cylinder.
At Vedantu, we simplify topics like Surface and Lateral Area of 3D Figures: Prisms so you can focus on building your maths skills for school, entrance exams, and modern careers.
In summary, understanding how to calculate the surface and lateral areas of prisms enables students to solve practical and academic problems confidently. Knowing when to use which formula and how to avoid common mistakes is key for excellent results in exams and real-life applications.
FAQs on Prism Surface Area and Lateral Area Formulas Explained
1. What is the formula for the lateral surface area of a prism?
The lateral surface area (LSA) of any prism is found by multiplying the perimeter of its base by its height. This is because the lateral faces, when unfolded, form a rectangle.
LSA = Perimeter × Height
2. How do you find the surface area of a 3D triangular prism?
To find the total surface area (TSA) of a triangular prism, you need two things: the lateral surface area and the area of its two triangular bases.
• Calculate the perimeter of the triangular base.
• Multiply that perimeter by the prism's height to get the LSA.
• Calculate the area of one triangular base.
• Double that area and add it to the LSA to get the TSA. TSA = LSA + 2 × (Area of triangular base)
3. How do you find the lateral area and surface area of each prism?
The method for calculating the lateral surface area (LSA) and total surface area (TSA) depends on the prism's shape. For any prism:
• LSA = Perimeter of base × Height
• TSA = LSA + 2 × Area of base. Specific formulas exist for rectangular, triangular, and other types of prisms.
4. What is the formula for surface area in 3D?
There isn't one single formula for all 3D shapes. The formula for surface area varies depending on the shape. For prisms, the total surface area (TSA) is calculated as TSA = LSA + 2B, where LSA is the lateral surface area and B is the area of the base. Other shapes (cubes, cuboids, cylinders, cones, spheres) have their own unique surface area formulas.
5. What is the formula for the total surface area of a prism?
The total surface area (TSA) of a prism is the sum of its lateral surface area (LSA) and twice the area of its base (B). The formula is: TSA = LSA + 2B. Remember, LSA = perimeter of base × height.
6. What is the lateral surface area of a prism?
The lateral surface area (LSA) of a prism is the area of all its sides, excluding the top and bottom bases. It's calculated by multiplying the perimeter of the base by the prism's height. LSA = Perimeter × Height.
7. How do you find the surface area of a triangular prism?
To calculate the surface area of a triangular prism, first find the area of the two triangular bases and the three rectangular sides. Then, add all these areas together. Alternatively, use the formula: TSA = LSA + 2B where LSA = (perimeter of triangle) × height; B = area of triangle.
8. What is the formula for total surface area of a rectangular prism?
A rectangular prism (or cuboid) has six rectangular faces. The total surface area (TSA) is given by: TSA = 2(lw + lh + wh), where 'l' is length, 'w' is width, and 'h' is height.
9. What is the difference between lateral surface area and total surface area?
The key difference lies in what surfaces are included. Lateral surface area (LSA) only considers the area of the sides of a 3D shape (excluding the bases). Total surface area (TSA) includes the areas of all faces, including the bases.
10. How are surface area formulas applied in real life?
Surface area calculations have many real-world applications. For example:
• Determining the amount of paint needed for a wall (rectangular prism)
• Calculating the material needed for packaging (various prism shapes)
• Estimating the heat loss from a building (considering exterior surface area)
• Designing efficient containers to minimize material waste (optimizing surface area)
11. Why is the perimeter of the base important in the lateral area formula?
The perimeter of the base is crucial in the lateral surface area (LSA) formula because it represents the total length of the sides of the prism that form the lateral faces. When you unfold the lateral faces, they form a rectangle with length equal to the base perimeter and width equal to the height.
12. How do surface area calculations change for oblique prisms?
For oblique prisms (where the lateral faces are not perpendicular to the bases), the lateral surface area (LSA) calculation is more complex. The simple formula (Perimeter × Height) no longer applies directly. You need to consider the individual areas of each parallelogram lateral face and sum them up.
